{
  "id": "1910.11290",
  "version": "v1",
  "published": "2019-10-24T17:06:35.000Z",
  "updated": "2019-10-24T17:06:35.000Z",
  "title": "Post-Critically Finite Maps on $\\mathbb{P}^n$ for $n\\ge2$ are Sparse",
  "authors": [
    "Patrick Ingram",
    "Rohini Ramadas",
    "Joseph H. Silverman"
  ],
  "comment": "32 pages",
  "categories": [
    "math.DS",
    "math.AG"
  ],
  "abstract": "Let $f:{\\mathbb P}^n\\to{\\mathbb P}^n$ be a morphism of degree $d\\ge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $k\\ge1$ and $\\ell\\ge0$ such that the critical locus $\\operatorname{Crit}_f$ satisfies $f^{k+\\ell}(\\operatorname{Crit}_f)\\subseteq{f^\\ell(\\operatorname{Crit}_f)}$. The smallest such $\\ell$ is called the tail-length. We prove that for $d\\ge3$ and $n\\ge2$, the set of PCF maps $f$ with tail-length at most $2$ is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with $\\ell=0$, are not Zariski dense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-24T17:06:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "37F10",
      "37F45",
      "37P45"
    ],
    "keywords": [
      "post-critically finite maps",
      "zariski dense",
      "critical locus",
      "parameter space",
      "periodic critical loci"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}